ON SEQUENCE SPACES EQUATIONS OF THE FORM ET + Fx = Fb FOR SOME TRIANGLE T

Given any sequence a = (an)n≥1 of positive real numbers and any set e of complex sequences, we write ea for the set of all sequences y = (yn)n≥1 such that y/a = (yn/an)n≥1 є e; in particular, sa (c) denotes the set of all sequences y such that y/a converges. we denote by w∞ and w0 the sets of all sequences y such that supn( n^-1∑n k=1 |yk|) < ∞ and limn→ ∞ ( n-1∑n k=1 |yk|) = 0. we also use the sets of analytic and entire sequences denoted by λ and γ and defined by supn|yn|1/n < ∞ and limn→ ∞ |yn|1/n = 0, respectively. in this paper we explicitly calculate the solutions of (sse) of the form et+fx = fb in each of the cases e = c0, c,(ell)∞ ,(ell)p, (p≥1), w0, γ, or λ, f = c, or(ell)∞, and t is either of the triangles∆, or ∑, where∆ is the operator of the first difference, and ∑ is the operator defined by ∑ny = ∑n k=1 yk. for instance the solvability of the (sse) γ∑+λx = λb consists in determining the set of all positive sequences x = (xn)n that satisfy the statement: supn n{(|yn|/bn)1/n} < ∞ if and only if there are u, v єw with y = u+v such that limn→ ∞│∑n k=1 uk│1/n = 0 and sup n {( |vn| /xn)1/n } < ∞ for all y.